TY - JOUR

T1 - An Alberti's rank-one theorem for martingales

AU - Ayoush, Rami

AU - Столяров, Дмитрий Михайлович

AU - Wojciechowski, Michał

PY - 2025

Y1 - 2025

N2 - We prove that the polar decomposition of the singular part of a vector measure depends on its conditional expectations computed with respect to the q-regular filtration. This dependency is governed by a martingale analog of the so-called wave cone, which naturally corresponds to the result of De Philippis and Rindler about fine properties of PDE-constrained vector measures. As a corollary we obtain a martingale version of Alberti’s rank-one theorem.

AB - We prove that the polar decomposition of the singular part of a vector measure depends on its conditional expectations computed with respect to the q-regular filtration. This dependency is governed by a martingale analog of the so-called wave cone, which naturally corresponds to the result of De Philippis and Rindler about fine properties of PDE-constrained vector measures. As a corollary we obtain a martingale version of Alberti’s rank-one theorem.

KW - martingale

KW - rank-one property

KW - vector measure

UR - https://www.mendeley.com/catalogue/257836b9-e8a0-37e9-bc60-cd9462334627/

U2 - 10.2140/pjm.2025.334.1

DO - 10.2140/pjm.2025.334.1

M3 - Article

VL - 334

JO - Pacific Journal of Mathematics

JF - Pacific Journal of Mathematics

SN - 0030-8730

IS - 1

ER -